Endpoint Sobolev regularity of higher order maximal commutators

This paper is devoted to presenting some W-1,W-1-regularity properties of higher order maximal commutator and its fractional variant. More precisely, let k >= 1, alpha. [0, 1) and b is an element of L-loc(1)(R). We consider the following k-th order fractional maximal commutator m(b,alpha)(k) f(x) = sup(2t)t>0(alpha-1) integral(x+t)(x-t) |b(x) - b(z)|(k)| f(z) | dz, x is an element of R, which includes the k-th order maximal commutator m(b)(k), corresponding to the critical case alpha = 0. We establish the boundedness and continuity of the map f (sic) (m(b,alpha)(k) f)' from W-1,W-1( R) to L-q (R), provided that alpha is an element of [0, 1), q is an element of (1,infinity), b is an element of Lip(R) and b' is an element of L-1(R). We emphasize that our work not only improves essentially some known results, but also provides a new and simpler proof of some known ones. It should be also pointed out that the above results are new, even in the special case k = 1.